Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined action of $G$ on structure sheaf $\mathcal{O}_X$ such that $\mathcal{O}_{X/G}= \mathcal{O}_X^G$).
Denote by $p: X \to X/G$ the induced well defined projection morphism.
Firsly I heard that in this case $p$ is called a "Galois morphism". Why?
Now consider the exact sequence
$$0\rightarrow \mathcal{G} '\rightarrow \mathcal{G} \rightarrow \mathcal{G}''\rightarrow 0$$
of coherent sheaves over $X$.
Let assume that after applying the pushforward /direct image functor $p_*$ the sequence stay exact.
My second question is if there exist and how defined if exist an induced (canonical?) action of $G$ on any "pushed forward" coheherent sheaf $p_*\mathcal{F}$
My intention is the following: If the last question has a positive answer then I can apply to the exact sequence
$$0\rightarrow p_*\mathcal{G} '\rightarrow p_*\mathcal{G} \rightarrow p_*\mathcal{G}''\rightarrow 0$$
in this case the functor $^G$ on sheaves over $X/G$ via $p_*\mathcal{F} \to (p_*\mathcal{F})^G$
Then I'm looking for criterions which garantee the exactness of this functor.